# How would show that points (4,2),(3,3),(2,4) are on the same line?

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### 3 Answers

Another way of proving (4,2), (3,3) and (2,4) are on the same line is to derive the equation of the line passing through any two of the points and showing that the third point lies on the line.

The equation of the line passing through (4, 2) and (3, 3) is `(y - 2)/(x - 4) = (3 - 2)/(3 - 4)`

=> `(y - 2)/(x - 4) = -1`

=> y - 2 = 4 - x

=> y + x - 6 = 0

Substituting the coordinates of the point (2, 4) in the equation gives 4 + 2 - 6 = 0.

**This shows that the points (4,2), (3,3) and (2,4) are on the same line.**

Another way is to show that the slope between any 2 given points is the same.

slope = `"rise"/"run" = (y_2 - y_1)/(x_2-x_1)`

Between (4, 2) and (3, 3) = `(2-3)/(4-3) = -1/1 = -1`

Between (3, 3) and (2, 4) = `(3-4)/(3-2) = -1/1 = -1`

Between (4, 2) and (2, 4) = `(2 - 4)/(4 - 2) = -2/2 = -1`

**The slope is the same between any 2 given points, therefore all 3 points lie on the same line.**

You may check the collinearity of the given points using the following determinant formula, such that:

`Delta = [(4,2,1),(3,3,1),(2,4,1)]`

You need to notice that each of the three rows of determinant contains the coordinates of the given points. Since the problem provides two coordinates points, the last position, on each row, is filled in with 1.

You need to evaluate the determinant, such that:

`Delta = [(4,2,1),(3,3,1),(2,4,1)]`

`Delta = 12 + 12 + 4 - 6 - 16 - 6 = 0`

Since `Delta = 0` yields that the given points are collinear.

**Hence, testing the collinearity of the given points, using determinant formula, yields that the points are collinear.**

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